Lemma 59.31.1 (04HV)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.31: Supports of abelian sheaves
Lemma 59.31.1 (cite)

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Lemma 59.31.1. Let $S$ be a scheme. Let $\mathcal{F}$ be a subsheaf of the final object of the étale topos of $S$ (see Sites, Example 7.10.2). Then there exists a unique open $W \subset S$ such that $\mathcal{F} = h_ W$.

Proof. The condition means that $\mathcal{F}(U)$ is a singleton or empty for all $\varphi : U \to S$ in $\mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$. In particular local sections always glue. If $\mathcal{F}(U) \not= \emptyset $, then $\mathcal{F}(\varphi (U)) \not= \emptyset $ because $\{ \varphi : U \to \varphi (U)\} $ is a covering. Hence we can take $W = \bigcup _{\varphi : U \to S, \mathcal{F}(U) \not= \emptyset } \varphi (U)$. $\square$

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